the ultimate shape of pebbles, natural and artificial

The Ultimate Shape of Pebbles, Natural and ArtificialAuthor(s): Lord RayleighSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 181, No. 985 (Dec. 31, 1942), pp. 107-118Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/97802 .

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The ultimate shape of pebbles, natural and artificial

BY LORD RAYLEIGH, F.R.S.

(Received 16 July 1942)

[Plates 1, 2]

Flint pebbles that have been worn down enough to be symmetrical may be spherical, or may approximate to prolate or oblate spheroids or to ellipsoids. When, however, the section is at all elongated, it is not as a rule accurately elliptical, but except at the axial points it lies entirely outside an ellipse adjusted to the same axes. Thus, if one of the axes is much smaller than the other, the pebble is much flatter than an ellipsoid.

Considering the quasi-spheroidal pebbles, whether prolate or oblate, these are always flattened at the poles, and the very oblate ones become tabular or even concave at the poles. These statements hold for flint pebbles, but a large quartzite pebble is figured which is pretty accurately an oblate spheroid, the meridional sections being truly elliptical.

Experiments are described with chalk pebbles, initially shaped as prolate or oblate spheroids, and the change of figure under abrasion is observed. This depends in some degree on what is the abrasive. Steel nuts, nails ('tin tacks') and small shot were used. In general the axes tend to approach equality, but not rapidly enough for the spherical form to be attained before the pebble has disappeared. The form initially spheroidal becomes flattened at the poles just like the natural flint pebbles, and may become concave, as flints sometimes do.

The circumstances determining the rate of abrasion at any point are considered, and it is shown that this abrasion cannot be merely a function of the local specific curvature. The figure at points other than the one under consideration comes into question, so that the problem in this form has no determinate answer. It is shown by simple mechanical con- siderations how the concave form arises.

The large majority of pebbles found in nature though rounded in outline are of unsymmetrical shape. This is no doubt mainly due to their not having undergone enough attrition to attain symmetry, but also in some cases to the material not being homogeneous-for it is sometimes found in experiment that the original artificially imposed symmetry is lost in the course of attrition. The present paper only deals with that minority of natural pebbles which are symmetrical, and with the experimental study of attrition, starting with definite symmetrical shapes.

My own recent observations of natural pebbles have been chiefly made on the flint deposits classed as glacial drifts, the beaches not being now accessible. Spherical pebbles are occasionally met with, but they are so rare in my district (Terling, Essex, not far from Chelmsford) that they are sometimes put aside as minor curiosities. I have found or been given five or six of them in the course of a year's interest in the subject, in a district where the gravel is extensively put down on roads of occupation, farmyards and the like. Pebbles approximating to oblate spheroids are fairly common, especially in small sizes, 2 to 3 cm. in diameter. Prolate forms of revolution are very rare, indeed practically non-existent. Long finger-like natural shapes' of flint are not uncommon in the chalk and in the drifts, and it might be imagined that they wouldi yield approximate prolate spheroids by attrition, but in fact elongated pebbles when found are practically always wanting

Vol. r8r. A. (3I December I942) [ 107 ] 8



108 Lord Rayleigh

in symmetry. It would seem that this shape is unstable, and loses symmetry instead of gaining it in course of attrition. Experimental observations casually made tend to confirm this view.

Coming now to a closer view of the exact shape, it is often found that the spheres are fairly accurate. I have seen a spherical quartzite pebble from Woolshed, Beachworth, Victoria, Australia that was almost as accurately shaped as a billiard ball. It would seem that if what may be loosely described as the three principal axes are not very different initially (cube, e.g.) then the spherical form is con- tinually approached as attrition proceeds. Oblate forms very close to the sphere are certainly not often met with. I have, in fact, never found one with the polar axis as much as 07 of the equatorial diameter.

The principal sections of a pebble, if not circular, are seldom accurately elliptical. An example is seen in figure 1 (plate 1), which represents the section ac of a fairly symmetrical pebble having its axes 2a = 2-95 cm., 2b = 2*12 cm., 2c = 1'49 cm. The section was drawn by projecting the enlarged silhouette of the pebble optically, and going round the shadow with a pencil. An ellipse was drawn with the same axes 2a and 2c. It will be seen that the ellipse lies within the pebble section, except at the four points on the axes, when it was adjusted to fit. The figure shown was reduced photographically from the enlarged graph.

This inscribed ellipse was drawn as the simplest way of showing that the section was not elliptical. The circumscribed ellipse which could only be drawn by tenta- tive methods would show more directly how the actual pebble could be derived by local abrasion or figuring (or rather disfiguring) of an ellipsoid.

Pebbles like this one, having three unequal axes, exhibit a flattened shape in each of the principal sections, the curve bulging away at four places from the inscribed ellipse, and naturally this is more apparent the farther the section departs from a circular form. Whatever the significance of the flattened form, it is most simply exemplified when one section is circular, so that no unnecessary complica- tions arise.

Such pebbles then- are usually flattened at the poles* and in the case of flints they are invariably flattened, so far as my experience goes.

The following photographs are silhouettes (shadows cast by a distant arc lamp) in actual size, the axis of the figure being in each case vertical. Figure 2 (plate 1) is the nearest approach to a prolate figure of revolution that I have been able to find. The axial measurements are 3-18 cm., 2-21 cm., 1*71 cm. It is obvious by inspection that the ends are much blunted compared with the circumscribed ellipse.

Figure 3 (plate 1) is an oblate figure of revolution, again much flattened in the axial direction.

There exist, however, oblate pebbles, which are not flattened in this way, but approximate closely to oblate spheroids. Figure 4 (plate 1) shows a pebble of

* In popular language the earth is said to be flattened at the poles. This means flattened from the sphere to the oblate spheroid. The pebble is flattened beyond this, the section being no longer elliptical.



The ultimate shape of pebbles, natural and artifical 109

quartzite which I owe to the kindness of my friend, Sir D'Arey Thompson, F.R.S. He was unable to say with certainty where he collected it, but thinks it most probable that it was on the banks of the River Spey in Inverness-shire. Figure 5 (plate 1) is an ellipse of nearly the same dimensions as the pebble section for comparison. In the same collection were two other similarly shaped pebbles of some other fine-grained igneous or metamorphic rock, not precisely identified.

Returning to the oblate flint pebbles, it is found that some of the thinnest ones are not merely flat at the poles, but actually concave. Figure. 6 (plate 1) illustrates this. The concavity cannot be accurately photographed in profile, for obvious geometrical reasons, but a straight-edge has been placed over the concave surface, and with suitably adjusted positions of light and camera lens scattered light passes below the straight-edge, proving concavity. The other side is also concave though the illumination cannot well be arranged to show this in the same picture. The pebble measures 3*0 x 2*2 x O08cm., andthe radii of concave curvature are estimated at 19 cm. and 30 cm.

In the case of natural pebbles there are no means of knowing definitely by what stages the individual pebble has attained its present shape. In particular we cannot say a priori whether a flattened surface has not been formed by the sides having been embedded, and the face alone exposed to attrition. It is true that in the case of a symmetrical pebble, we should have to assume that the stone had been turned over and the other side similarly flattened in its turn: but thi' cannot be clearly excluded.

Here comes in one great advantage of experimental conditions. They can be so arranged that no accident can have any preponderating influence and the influences at work are those which the statistical conditions themselves dictate. Further, the changing shapes of one particular pebble can be examined as attrition proceeds. Under natural conditions this cannot be done, and there is no definite information as to the shape which preceded that actually observed.

Many experiments have been made in the past on, the formation of pebbles by abrasion, and some references are given at the end of this paper (Wentworth I9I9,

Showe I932, Bullows I939, Krumhein I94I). Some of the investigators are chiefly interested in the question of what distance the-pebble must travel under water to produce a well-rounded form in the various kinds of material, and generally to see what can be inferred about the geological history of similar pebbles occurring in nature. I have not been able to find much about the questions here discussed, though it is of course widely recognized that the typical form of pebbles is flattened, and also that spheres are rare, and are not produced unless the original pieces have their three principal axes (using the terms in a loose general sense) nearly equal. It may be remarked that experiments on slate, or indeed any rock which has distinctive planes of cleavage or bedding, are not helpful in the present .discussion.

The method of investigation used is to begin with pebbles of a definite geometrical form, free from sharp edges, and to observe the modification of shape as



110 Lord Rayleigh

attrition proceeds. There is little choice as to the initial form. An ellipsoid is clearly indicated. If we were to go beyond this we should have to consider surfaces of the fourth degree, which is scarcely practicable.

It is useful to consider in the first instance what relation must hold between the specific curvature at any point of an ellipsoid and the linear rate of attrition measured perpendicular to the surface in order that the shape should be unaltered by the process of attrition.

A general discussion of the curvature of surfaces is given, e.g. by Thomson & Tait (I879, p. 101). The reader may be reminded that at any point of a surface there are two principal curvatures. These are the curvatures of plane sections normal to the surface, these sections being chosen so as to have maximum and minimum curvature. It may be shown that they are mutually perpendicular. The specific curvature of the surface can be calculated as the product of these two principal curvatures.

In the particular case of the ends of the axes of an ellipsoid the specific curvatures are easily found. If a, b, c are the semi-axes, the specific curvatures are

a2/b2c2, b2/a2c2 c2/a2b2.

Multiplying through by a2b2c2, we see that these are in the ratio

a4:b4:c4.

If, therefore, in any given case the ratios of the axes a: b: c are found to be unaltered by abrasion, it follows that in this case the rate of abrasion at their ends must be as the fourth roots of the specific curvatures at these ends. It can be shown that the same holds at other points of the ellipsoid though naturally the geometry in this general case is not quite so simple. (See appendix at the end of the paper.)

The questions of special interest in connexion with the abrasion of ellipsoids appear to be all sufficiently exemplified by prolate and oblate spheroids, which are much easier to shape in the first instance and to examine and discuss afterwards. Accordingly these have been used exclusively in the experimental work.

Chalk was used as the material because it is soft. This makes it easy to shape the spheroids in the first place, and the desired amount of abrasion can be carried out in periods of at most a few hours, and often less. The method of producing chalk spheroids is simple. An elliptic aperture is made in a piece of sheet brass about 1 mm. in thickness. This would best be done with an oval chuck, as used in orna- mental turning. In the absence of such a tool, the ellipse may be scribed on the brass, which can be done well enough for the purpose by the common method of a loop of (linen) thread round two drawing pins. These pass through holes in the brass into a board below. The aperture is then filed out up to this mark. Having prepared the elliptical template, a piece of chalk is sawn and rasped to near the desired shape, and then revolved in the fingers against the template used as a cutting tool until it will just go through. In a few minutes a very satisfying



The ulttimate shape of pebbles, natural and artificial 111

spheroid can be produced. The same template can of course be used to make a prolate or oblate spheroid of a given ratio of axes. The chalk used was from the Cherry Hinton pit near Cambridge. It occasionally contains shells or other hard inclusions which resist abrasion more than the body of the material, and produce local irregularities, but the trouble from this cause is not serious. The abrasion was

0 9 _ 0 o

0.8 0.8 C

0.7 -0.7 -

O* 6 - 0 O6- .4)

0 -'3V.

o.s !: o.sLX

+ C

0

0.3 -0.3-

0.2 *0.2

0 1 0 1~

olI I AI I

0 01 02 03 04 0 01 02 03 04 05 0.6 loss of diameter (cm.) loss of diameter (cm.)

FIGURE 7. Prolate. FIGURE 8. Prolate.

carried out in the absence of water. This is a departure from the conditions which generally prevail in the formation of natural flint pebbles, but it may be doubted whether using the chalk wet would really imitate these conditions more closely, for unlike flint, chalk gets soft and muddy under water. In fact, wet chalk is too soft for satisfactory measurement as attrition proceeds and to stop and dry it on each occasion would seriously delay the work. Further it is found that patches of local softness leading to irregular figure are more noticeable when the material is wet.



112 Lord Rayleigh

The shaped chalk pebble was placed in a metal box measuring 18 x 10 x 8 cm. with either 470 g. of steel hexagon nuts somewhat various but weighing on the average 7*5 g. each; or 420 g. of small nails ('tin tacks'), length 10 mm., weight 0 3 g.; or 1200 g. of small shot, diameter of pellets 2*5 mm., weight 0-1 g. There was nothing special about these quantities, which were originally taken at random, and adhered to.

0.6

04 -

0.3

O*/

0.2

0 0.1 0.2 0.3 0_4 O 06

loss of diameter (cm.)

FiGURE 10. Spherite.

o /-

0 0 2 _

0 0.1 0.2 0.3 0-4 0-5 0.6 loss of diameter (cm.)

FIGURE I9. Oblaeria.



The ultimate shape of pebbles, natural and artificial 113

The box rotated in a lathe about a central axis parallel to the intermediate dimension, and the contents fell from one end to the other about once a second. The chalk pebble was taken out from time to time, and measured axially, and along three or (preferably) two equatorial diameters, by sliding calipers.

In some experiments chalk spheres of vaTious sizes and chalk spheroids were abraded together for comparison, but no further reference is made to these tests in the present paper.

If a glass lid is placed on the tin box it becomes possible to observe the pebble during attrition, and to see if there is any marked tendency for it to assume one presentation or another towards the stream of impinging abrasive. So far as could be seen, there was no marked tendency of this kind, whether the pebble was prolate or oblate.

The experimental results are plotted on the diagrams, figures 7-12. It will be observed that the losses of length (polar) and diameter (equatorial) are plotted, instead of the actual lengths and diameters. These losses have to be determined by the difference of much larger quantities, which is disadvantageous, and the uncertainty (say, 0-02 cm.) is appreciable on the scale of the diagrams. The irregularities could be smoothed out by making adjustments of this order of magnitude.

0-4

03

0 01 0-2 0-3 04 O5 06 0Q7 08

loss of diameter (cm.)

FIGURE 1 1. Oblate.

The diagrams are arranged in order, with the loss of diameter plotted as abscissa, and the loss of length as ordinate. Thus the axis of figure of the spheroid is to be thought of as vertical in each diagram as generally throughout this paper. The spheroids pass from prolate through spherical to oblate along the series of diagrams; the sections inset in each skow these in comparative dimensions. On each diagram there is a point- marked thus x . The abscissa of this point is (on the scale of the diagram) 1/10 the original diameter of the spheroid, and its ordinate 1/10 given on the original length.- Joining this point to the origin we should get a line whose



114 Lord Rayleigh

slope indicates the original ratio of length to diameter. (The line is not actually drawn because it would tend to overcrowd some of the diagrams.) I refer to it as the standard line.

If the ratio length/diameter were unaltered by abrasion, the graphs would be straight lines coincident with the standard line, the cross x lying on this line into which the graphs all coalesce. This would of course be actually realized for the sphere, the standard line being at 45?. The spherical case is indicated in figure 8, which merely expresses the fact that a sphere remains a sphere whatever kind of abrasive materials are used. This has often been incidentally verified in the present work, though verification is hardly needed. The only object of figure 9 is to show the relation with other cases at a glance.

It will be observed that in cer-tain of these other cases the ratio length/diameter remains unaltered within the limits of experimental error. This applies, e.g. to prolate spheroids with the ratio length/diameter =about 1*4 when the abrasion is by steel nuts (figure 8).

It also applies to oblate spheroids, length/diameter about 0.5, abraded by nails (figure 11).

In all remaining cases shown on the diagrams the abrasion definitely disturbs the ratio of axes and for prolate spheroids the ratio length/diameter is diminished. In other words the prolate spheroid either maintains its shape, or is shortened towards the spheroid form.

An oblate spheroid may either: (1) Maintain its ratio of length/diameter as in figure 11 (nails), (2) or (more commonly) the ratio length/diameter may increase, tending towards

the sphere, (3) or the ratio length/diameter may decrease, as in figure 11 (shot), length/

diameter 05. This is a case unique in the diagrams of a tendency to change away from the sphere, the original oblate form becoming more oblate.

The changes of length and diameter are readily measured experimentally, and give a good deal of information: but clearly they cannot decide whether the figure remains spheroidal or not. For this purpose the projected shadow of the specimen must be examined.

The profile photographs (figures 13, 14a, 15, 16, 17, 18, 19, plate 2) illustrate the original shape (outer margin) and the final shape (inner margin). Figures 14b and c show the components of figure 14a separated. The composite photographs were made as follows. The specimen was tacked to a glass plate with sealing wax, and the photographic plate placed in contact with this glass plate. A distant arc lamp threw a sharp shadow of practically actual size. A similar photograph of the brass stencil used to shape the spheroid originally was made. Positives were printed by contact on glass from these two negatives, assembled film to film in the proper relative positions, and fastened together with sealing wax. The pictures here given were printed from these assemblages.

(It was at first attempted to adjust the chalk pebble in the centre of the stencil,




and to take the original picture at one exposure, but a satisfactory adjustment was not easily attainable in this way.)

It appears very clearly from these photographs that the figure does not in general remain spheroidal. A spheroid, whether prolate or oblate, is flattened at the poles away from the spheroidal form, as in the case of the natural pebbles in figures 2 and 3. This is evident by inspection of the photographs, and has been verified more objectively by the method illustrated in figure 1, though this is hardly necessary. Figures 13, 14 and 15 are prolate figures. The remainder are oblate.

The flattening of the prolate spheroid is at the place where the initial curvature and rate of abrasion is greatest. The flattening of the oblate spheroid is at the place where the initial curvature is least, and the linear rate of abrasion small. (It is to be noticed however that this rate of abrasion is not the minimum, which is at an intermediate latitude. Its precise location is difficult.)

0.3 _ w

i 01 s ho

0 0*1 02 0*3 04 05 0-6 07 08 09 loss of diameter (cm.)

FIGURE 12. Oblate.

Consider the case where the ratio of the axes is preserved unaltered as in figure 13 (see also figures 8, 11, 12). In such a case, as we have seen, the abrasions at the pole and at the equator must be as the fourth roots of the initial specific curvatures. If the same held everywhere, the spheroidal form would be preserved.

As it is not preserved, we must conclude that abrasion is not a function of the specific curvature only. The same follows from the fact that starting from a spheroid of given shape and therefore given curvatures at the poles and the equator, we find in general a different ratio of polar to equatorial abrasion according to what abrasive is used-nails, steel nuts, or small shot. It is therefore useless to propose an amended law connecting the specific curvature with the linear abrasion at that point, for no such law can hold good.

This raises the question what the linear abrasion at a point on the surface does depend upon, if it is not determined by the specific curvature there. The considera- tions above given about the different abrasives show equally that it cannot be fixed by any other function of the principal curvatures, their sum for example. What then remains ? The edges of a rectangular block are quickly rounded, which shows that a large local principal curvature (as distinguished from the specific



116 Lord Rayleigh

curvature) has an important influence, but it cannot be the only one. The general outline of the specimen must have an effect in guiding the motion of the abrading bodies large or small in its neighbourhood, and hence the shape at points other than the one under consideration must be important. - In most cases it would be difficult to attempt any further analysis; but there is a special case which lends itself to the attempt. This is the case of a flat surface with sharp edges. If abrasion depended solely on curvature, the flat surface ought not to be abraded at all; but common sense rejects such a conclusion, and this alone makes it evident that curvature is not everything.

Experiments on flat surfaces in fact lead to a very curious result. If we reduce an oblate ellipsoid sufficiently we get an approximately flat surface as in figure 19. After this amount of reduction, the specimen of chalk becomes somewhat rough and irregular though it shows a tendency to concavity. To pursue the matter further, it was found desirable to start with a flat surface, and to use steatite rather than chalk because it is of more uniform hardness and does not develop these local irregularities.* Starting with a flat disk of steatite with or without rounded edges, and abrading it with nails or steel nuts, a definite concavity developed on the initially flat faces, and shows conspicuously when tested with a straight-edge. Tests made with templates indicate that the radius of curvature of the concavity is of the order of 20 cm. This is seen in the photograph, figure 20, where a straight-edge placed over the surface shows the concavity by diffused light passing below, as with the concave natural pebble, figure 6. This concavity can be equally well obtained using as abrasive rounded flint pebbles instead of the steel nuts; it has therefore no necessary connexion with the angularity of the latter. On the other hand it is not obtained with small shot as the abrasive. We may therefore base the interpretation on assuming that the abrasive masses are com- parable in size with the specimen.

-a . b

FIGURE 21 a, b. Illustrating formations of a concave surface.

Suppose now that an abrasive mass approaches the specimen, and let its path be such that the centre lies beyond the edge at the point of impact (figure 21a). The edge will be rounded, and the abrasive mass will be deflected away, and will not produce any effect on the flat surface near the edge. If, on the other hand, the centre of abrasive mass strikes definitely inside the edge (figure 21 b) it will dig

* Probably steatite would have been a better material to use throughout, but I was for a time unable to procure it.




a hole in the flat surface at the point of impact. Thus, it is fairly easy to see that the statistical result will be a rounding of the edge, and the digging of a concavity in the flat surface.

It will be noticed that this explanation requires the impinging abrasive mass to be of considerable size. The explanation applies whether the impact is normal or moderately oblique.

It is believed that flint pebbles such as figure 6, after passing the flat stage have become concave in this way. It should be remarked that concave flint surfaces can originate by fracture, and the general shape and patination sometimes indicate that this is in fact their origin. There is no absolutely certain criterion for deciding, but the indications are that a flat biconcave specimen like figure 6 has been abraded concave.

In the experiments on abrasion which have been described, the abrasion has not been carried to great lengths, the object being rather to study the initial rate of abrasion of a spheroid. The tendency is usually towards a sphere, but it is so slow that the pebble will in most cases have disappeared before the sphere is attained. A few experiments have been made when the attrition was carried to great lengths. Thus, a rectangular block of chalk was cut measuring 2-5 x 2-5 x 1-6 cm. and was ground down using steel nuts as the abrasive. The following results were obtained:

weight, g. approximate shape length/diameter 16-3 rectangular 0-694 4'0 oblate spheroid 0.71 009 oblate spheroid 0 74

Reducing the weight 183-fold has only achieved small progress towards the sphere. Some tests were made on a prolate ellipsoid of chalk, abraded by steel nuts:

length diameter cm. cm. ratio 3'66 2'92 1-26 2'51 1'96 1*28

It will be seen that over this range there is little change of relative dimensions, and what tendency there is is away from the sphere. When the abrasion was carried further, the circular symmetry was lost, the form becoming ellipsoidal rather than spheroidal. If this is typical it accounts for the rarity of prolate pebbles of circular symmetry.

We do occasionally find spherical pebbles however, and they are often rather good approximations to a perfect sphere. It would seem therefore that when a certain approximation to a sphere has been attained, progress towards equality must be much more rapid than the above tests would suggest -as likely. There is much more to be done in this direction, and materials harder and more homogeneous than chalk should be tried.



118 Lord Rayleigh

APPENDIX

Proof that if an ellipsoid is found to be reduced by abrasion-without change of shape, the thickness removed at any point P must be as the fourth root of the specific curvature at P.

Consider the quadrics confocal with the ellipsoid, and passing through P. One of these will be a hyperboloid of one sheet, the other a hyperboloid of two sheets. Let the semi-axes co-linear with the semi-axis a of the ellipsoid be a' and a" respectively. Let p be the perpendicular from the centre on to the tangent plane at P.

Then we have -(Salmon 1914, p. 171)

- a2b2c2 -12 1) 2= - I (as-a'2) (a2-a"2)

The principal curvatures at the point P are (Salmon I914, p. 159)

p/(a - a'2) and p/(a2 - a"2)

Thus for the specific curvature we have

p2/(a2 - a'2) (a2 - a'2) or a2b2c2/(a2 - a'2)2 (a2 - a/2)2.

The ratio of this to the specific curvature at the end of the semi-axis a is

a2b2c2 a2 b4c4 (a2 - a'2)2 (a2 - a"2) b2 o 2 a'2)2 (a2 _a2)2

If the material removed by abrasion -at P has the thickness 6p then, since similarity is assumed to be preserved, the ratio of this to the abrasion da at the end of a is given by

6p p 1 abc bc 7a a a (a2- a'2) _ (a2 a2) (a2 a2) (a2 -al/2)

This is the fourth root of the ratio of specific curvatures above found.

REFERENCES

Bullows, W. L. I939 Quarry Manager's J. 21, 277-279. Krumhein, W. C. 194I J. Geol. 44, 482-520. Salmon, G. I9I4 Geometry of three dimensions, 6th ed. London and Dublin. Showe, W. H. I932 Am. J. Sci. 224, 111. Thomson, W. & Tait, P. G. I879 Treatise on natural philosophy. Wentworth, C. K. I919 J. Geol. 27, 507.



Rayleigh Proc. Roy. Soc., A, volume 181, plate 1

1 4

2 5

3 6

FIGURES 1-6. Natural pebbles. Actual size. Shown in section, axis of figure vertical.

FIGURE 1. The ac section of a quasi-ellipsoidal pebble. Ellipse inscribed. FIGURE 2. Prolate quasi-spheroidal flint pebble. Axis vertical. FIGURE 3. Oblate quasi-spheroidal flint pebble. Axis vertical. FIGURE 4. Oblate spheroidal quartzite pebble. Axis vertical. FIGURE 5. Ellipse of nearly the same dimension as figure 4, for comparison. FIGURE 6. Biconcave pebble of flint. A straight-edge placed over it. Diffused light passes below the edge.

(Facing p. 118)



Rayleigh Proc. Roy. Soc., A, volume 181, plate 2

13 14a 14b 14c

15 16

17 18

19 20

FIGURES 13-19. Experimental pebbles, spheroids of chalk. Outside shows initial outline, inside shows final outline. Axis vertical. Actual size.

FiGURE 13. Prolate. Abraded with nails. FIGURE 14a. Prolate. Abraded with small shot. FIGURE 14 b, c. The same, with the initial and final figures separated. FIGURE 15. Prolate. Abraded with nails. FIGURE 16. Oblate. Abraded with nails. FIGURE 17. Oblate. Abraded with nails. FiC-URE 18. Oblate. Abraded with steel hexagon nuts. FIGURE 19. Oblate. Abraded with nails.

FIGURE 20. Steatite plane, abraded concave. Straight-edge applied as in figure 6.



the ultimate shape of pebbles, natural and artificial

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